↑

Properties

Label 2-3536-884.883-c0-0-1
Degree $2$
Conductor $3536$
Sign $1$
Analytic cond. $1.76469$
Root an. cond. $1.32841$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·3-s − 1.41·5-s + 1.00·9-s − 13-s + 2.00·15-s + 17-s − 2·19-s − 1.41·23-s + 1.00·25-s + 1.41·37-s + 1.41·39-s − 1.41·41-s − 1.41·45-s − 2·47-s + 49-s − 1.41·51-s + 2.82·57-s + 2·59-s + 1.41·65-s + 2.00·69-s + 1.41·73-s − 1.41·75-s + 1.41·79-s − 0.999·81-s − 1.41·85-s + 2.82·95-s − 1.41·97-s + ⋯
L(s)  = 1  − 1.41·3-s − 1.41·5-s + 1.00·9-s − 13-s + 2.00·15-s + 17-s − 2·19-s − 1.41·23-s + 1.00·25-s + 1.41·37-s + 1.41·39-s − 1.41·41-s − 1.41·45-s − 2·47-s + 49-s − 1.41·51-s + 2.82·57-s + 2·59-s + 1.41·65-s + 2.00·69-s + 1.41·73-s − 1.41·75-s + 1.41·79-s − 0.999·81-s − 1.41·85-s + 2.82·95-s − 1.41·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3536} (3535, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3277002579\)
\(L(\frac12)\) \(\approx\) \(0.3277002579\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + T \)
17 \( 1 - T \)
good3 \( 1 + 1.41T + T^{2} \)
5 \( 1 + 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
19 \( 1 + 2T + T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 2T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.41T + T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.384834271930976355628546370434, −8.061276129407222180130167502713, −7.15439334115265732764861487576, −6.51545815844012634608231443089, −5.76710309209004478077944344935, −4.88002582697107693856148440639, −4.30595309947897792128114646360, −3.51349153963213020333630223780, −2.14626577029987358917311339128, −0.50905375171074841774192792205, 0.50905375171074841774192792205, 2.14626577029987358917311339128, 3.51349153963213020333630223780, 4.30595309947897792128114646360, 4.88002582697107693856148440639, 5.76710309209004478077944344935, 6.51545815844012634608231443089, 7.15439334115265732764861487576, 8.061276129407222180130167502713, 8.384834271930976355628546370434

Graph of the $Z$-function along the critical line